Realising the Cognitive Potential of Children 5 to 7 with a Mathematics focus: Post-test and long-term Effects of a two-year intervention
Michael Shayer,
King’s College, London
Mundher Adhami
King’s College, London
Realising the Cognitive Potential of Children 5 to 7 with a Mathematics focus: Post-test and long-term Effects of a two-year intervention
Background. In the context of the British Government's policy directed on improving standards in schools, this paper presents research on the effects of a programme intended to promote the cognitive development of children in the first two years of Primary school (Y1 & 2, aged 5 to 7 years). The programme is based on earlier work dealing with classroom-based interventions with older children at both primary and secondary levels of schooling.
Aim. The hypothesis tested is that it is possible to increase the cognitive ability of children by assisting teachers toward that aim in the context of mathematics. A corollary hypothesis is that such an increase would result in an increase in long-term school achievement.
Sample. The participants were 8 teachers in one LEA and 10 teachers in another. Data was analysed on 275 children present at Year 1 Pre-test in 2002 and at long-term Key Stage 2 Post-test in 2008.
Method. Two intervention methods were employed: a Y1 set of interactive activities designed around Piagetian concrete operational schemata, and mathematics lessons in both Y1 and Y2 designed from a theory-base derived from both Piaget and Vygotsky.
Results. At post-test in 2004 the mean effect-sizes for cognitive development of the children—assessed by the Piagetian test Spatial Relations— were 0.71SD in one LEA and 0.60 SD in the other. Five classes achieved a median increase of 1.3 SD. The mean gains over Pre-test in 2002 for all children in Key Stage 1 English in 2004 were 0.51 SD, and at Key Stage 2 English in 2008—the long-term effect—were 0.36 SD, an improvement of 14 percentile points.
Conclusions. The main hypothesis was supported by the data on cognitive development. The corollary hypothesis is supported by the gains in English. The implications of this study are that relative intelligence can be increased and is not fixed, and that children can be led into collaborating with each other to the benefit of their own thinking, and that there does exist a theory-based methodology for the improvement of teaching.
Introduction
This paper presents research on the effects of a programme intended to promote the cognitive development of children in the first two years of Primary school (Y1 & 2, aged 5 to 7 years). Detail of its relation to previous intervention work with older children may be seen in Shayer (1999) and Shayer & Adhami (2007).
Earlier research on cognitive development of 10 to 16 year-olds had revealed that on Piagetian measures the levels of 11/12 year-olds entering secondary education spanned some twelve developmental years—i.e. from the average 6 year-old to the above-average 16 year-old (Shayer, Küchemann & Wylam, 1976). On the other hand research on 5 to 8 year-olds (Shayer, Demetriou & Pervez, 1988) had shown that 20% of the 7 year-olds were already at the mature concrete level (2B), the average level of 12 year-olds. A possible interpretation of these data —consistent also with data on 16 year-olds reported in Shayer & Wylam, (1978) and the development charted in Inhelder & Piaget, 1958—is that only 20% of the child population follow the complete developmental path described by Piaget. Bearing in mind that in Britain in 1945 only 20% of the population were deemed suitable for Grammar school education, it seemed desirable to test whether an intervention at the outset of Primary education might increase this proportion by the age of 7, thus increasing the potential for more children to benefit from the rest of their time in school. This intention was reinforced, after the start of the research project, by the data reported in Shayer, Ginsburg and Coe (2007) which showed that on the test of Piagetian conservations, Volume&Heaviness, the average performance of the 11/12 year-olds had slipped by 2004 to the level of the 8 to 9 year-olds in 1976.
The grounding of the RCPCM[1] work in the maths education research literature
The detailed specifics, and how they were used, are found in Shayer (2003) and Shayer & Adhami (2003). But from Cable (1997) and Vergnaud (1983) were drawn the importance of concepts of measure in even 5 year-olds’ development of their concept of number, neglected in much current teaching including the England and Wales National Numeracy scheme. From the work of Butterworth (1999) and Dehaene (1997)—and by implication from the early work of Dienes (1963)— come the importance of multiple representations of number using both sides of the brain. From Piaget & Szeminska (1952) comes the insight that even 5 and 6 year-olds have primitive concepts of multiplicative relations between numbers as well as additive, linking with measures, which are based on the ratio between the size of an object and the measure unit. And from much of the work of Nunes (2002; with Bryant, 1996)) detailed empirical research of how to promote multiplicative thinking was drawn on in the construction of TM lessons, with the intention, by the end of Year 2, of evening up the balance in children’s experience of number between the additive and multiplicative (ratio) aspects.
In Resnick, Bill & Lesgold (1992) one of the largest educational intervention effect-sizes ever achieved —2 SDs on mathematics achievement—is reported. The study describes work done in a single first year Primary class in arithmetic (6 to 7 year-olds): the authors comment
‘The children with whom we have worked come disproportionally from among the least favoured of American families. Many of them are considered to be educationally at risk…Yet these children learned effectively in a type of programme that, if present in schools at all, has been reserved for children judged able and talented–most often those from favoured social groups’.
This experiment, featuring a collaborative learning style for the children’s mathematics work having much in common with that used in the RCPCM classrooms, was also drawn on.
The Thinking Maths (TM) lessons
Insights into cognitive development from both Piaget and Vygotsky are used in the conduct of a TM lesson (Adhami, Shayer & Robertson, 2004; Adhami, Shayer & Twiss, 2005). Both believed in the virtues of collaborative learning. But Piaget himself made a seminal contribution to the mathematics literature in his book on Number (Piaget & Szeminska, 1952). From this and elsewhere (Shayer & Adey, 1981; Hart, 1981) it was possible to assess all the statements in the England and Wales National Curriculum (NC) in terms of demand levels from early concrete through to concrete generalisation. Similarly the levels of the different agendas in a TM lesson were also assessed. Given the data in the monograph Shayer, Demetriou & Pervez, 1988, the target aimed for in Year 2 (Y2) TM lessons, when the children would be 7, was mature concrete (for the Year 1 lessons it had been middle concrete). Each lesson started with an episode at a lower level, and the episodes—up to 3—were designed so that children from a wide range of initial cognitive abilities could each make some progress from their present level during each lesson.
Looking at the agenda of the TM lessons (and also the children) through Piagetian eyes is thus an essential aspect of the art. But Piaget’s work tells us little (except the importance of ‘cognitive conflict’) about how to teach the lessons. For this we turn to a view of Vygotsky’s concept of the Zone of Proximal Development (ZPD) that suggests that the mediating role of the teacher should be to maximize the collaborative processes through which the children can mediate each other’s learning (see the Appendix for an exposition of this view).
Each Episode of a TM lesson is a 3-Act play. The first, called Concrete Preparation, introduces the context of the lesson agenda at a level where all in the class can participate (5 to 8 minutes). Here the ‘argument within the collective’ is begun. Several children are asked to describe their views of what the task ahead involves, and to suggest possible strategies of solving it. In some cases the children are asked quickly in pairs, while sitting on the carpet, to attempt the easier parts of the task. Then different pairs are asked to offer their ideas, which others then discuss. In the second act, Construction, children, in groups from two to 4 or 5 are asked to go to their tables and attempt the first worksheet. In a few minutes time they will be asked to present their ideas to the rest of the class, so they do not waste time in ‘neat work’: they discuss vigorously with each other, using pencils on papers, until they feel they have something worth saying or showing (typically, 7 to 12 minutes). Act 3, Reflection, is again communal—often on the carpet in front of the whiteboard again—where their different solutions are shown and shared with the whole class (8 minutes at least). Thus in Acts 1 and 2 the community of learning is created, in which children offer different insights, which are made available for all to work with. But Act 3, well managed by the teacher, gives maximum chances for the individual ZPDs of the children to be completed as they witness the ideas produced by the other groups. Typically Act 3 leads naturally to the next Episode with agenda set usually at a higher level.
Method
The locations of the RCPCM research
The research team at Hammersmith and Fulham in 2001/2004: Hammersmith and Fulham (H&F) is a Local Education Authority (LEA) in Inner London, with some areas of social deprivation in which the experimental classes were located. The initial task in 2001/2002 was to create the TM lessons for Y1 use in 2002/2003. Four teachers, previously selected to assist in Professional Development (PD) for the CASE @ KS1 Let’s Think[2] lessons in H&F schools, were chosen as teacher-researchers. The research team consisted of the two authors from King’s College, an Advisory Teacher for H&F, and the four teacher-researchers, and met for a whole afternoon working session every 3 weeks. Initially the two authors would suggest mathematical contexts for a possible lesson, and then the whole team would work on how the lesson might be presented, and what worksheets etc. were needed. Then one of the teacher-researchers would offer to trial the lesson with a Y1 class, with the two authors in support. The first author would make detailed notes with a time line of the trial lesson, which would be written up for the next team afternoon. From that feedback the whole team would then revise the initial lesson plan, and one or the other of the authors would produce all the materials needed for the project teachers. Often an amended version of the lesson was re-trialled—sometimes more than once—before the team were satisfied with it. Later some of the ideas for the TM lessons came from the teacher-researchers. Twelve TM lessons were generated for 2002/2003 (Y1) and 19 for 2003/2004 (Y2). Thus the generation of lessons was based on the same processes of collaborative learning, with the teacher-researchers, as the project teachers would be asked to use with their children.
The research team at Bournemouth in 2001/2004: Bournemouth is a town on the south coast of England with a school intake about the National average. The H&F and King’s team had the responsibility of generating the TM lessons. But the Bournemouth LEA on the south coast of England offered to run a parallel replication of the RCPCM intervention in 2002/2004 in their schools—but taking the collaborative learning aspect of the classroom practice as their focus.
The research design: 2002 to 2004
In a recent research project CASE@KS1.H&F[3], in the Local Education Authority (LEA) of Hammersmith & Fulham, Adey, Robertson & Venville (2002) had reported for a one-year intervention with Y1 children (5/6 years of age) Post-test effect-sizes in relation to Control schools of 0.47 and 0.43 standard deviations (SDs) on Piagetian tests. Children were given, every week, interactive and collaborative learning focused on the major concrete operational schemata—classification, seriation, spatial perception, time relations and causality— described by Piaget. Their lessons were called Let’s Think (Adey, Robertson&Venville, 2001). Their interactive learning was conducted for some 20 to 30 minutes each day with groups of six children around a table with the teacher. Thus by the end of the week every child had experienced the activity (given 30 children in the class) and the teacher had both been able to mediate the children into collaborative learning, and also learnt much about their individual learning difficulties. Yet the exercises in thinking had not been located within the context of any specific part of the National Curriculum. This led to the design of the RCPCM Project[4] that the Hammersmith & Fulham LEA agreed to participate in.
In addition to experiencing the Let’s Think activities Year 1(Y1) children would also receive whole-class Thinking Maths (TM) lessons in order to focus their thinking skills within one major subject of the National Curriculum. Then in Y2, the second year of full-time education, the Let’s Think work would have ceased but the children, already used to the learning strategies practised in the Y1 TM lessons, would now receive further whole class TM lessons at a rate of about one every 10 days. In addition the teachers would be encouraged, where possible, to use the same teaching skills within the context of their ordinary Numeracy work, and also to establish a link between the children’s TM insights and the National Curriculum learning objectives.
The RCPCM intervention featured 8 experimental classes in 8 H&F schools and 10 experimental classes in 4 schools in Bournemouth (mean age at Y1 Pre-test 5 years, 7 months, SD 3.9 months). H&F provided 5 Control classes and Bournemouth provided 11. Table 1 shows the research design.
Table 1: RCPCM Main Study schedule
Year / Main Study Experimental Schools / Main Study Control SchoolsSept. 2002-July 2003 / Pre-tests: Piagetian Spatial Relations test
Y1 teachers use Let’s Think each week during the year
Y1 use 10 TM lessons during the year / Pre-tests: Piagetian Spatial Relations test
Sept. 2003-July 2004
July 2004 / Y2 teachers use 15 TM lessons during the year and also ‘bridge’ to their Numeracy work
Post-tests: Piagetian Spatial Relations test and KS1 SATs in Maths & English / Post-tests: Piagetian Spatial Relations test and KS1 SATs in Maths & English
The schools were selected by the LEA Advisory Teachers in consultation with their Heads, on the assumption that the whole year group would be involved. The H&F schools were all one-form entry: in Bournemouth all classes in the year group were involved. The selection was not random, but the samples are believed to be representative of the LEAs involved. In H&F the teachers received a two-day session of professional development (PD) twice each term in Y1 and Y2 up to and including the Spring term 2004. In Bournemouth there were four 1-day PD sessions from March to May 2002 on the Let’s Think teaching, seven 1-days of PD during Y1 for the Thinking Maths lessons, and two 1-day sessions in the first term of Y2.
Experimental and Control groups
Table 2 shows data used frequently in the UK as a measure of socio-economic status of families: the percentage of children in the schools allowed free school meals (%FSM). The control groups in each Local Education Authority (LEA) were chosen as far as possible to cover the range of intakes of the experimental groups. For the difference between the mean of all controls and all experimentals t = -1.8; p = 0.088 for %FSM. Thus although the socio-economic status of the intakes to the control schools is higher (%Free school meals are lower) the difference is not so great as to invalidate the study. It is convenient also to have control data that gives information on the higher levels of performance at post-test. Details of all schools are shown in Table 2 in order to display the wide variation in both experimental and control schools’ intakes.
Table 2: Socio-Economic data on the children in this study
Group / N / %Free School mealsH&F
Experimental group
1 / 29 / 20.1
2 / 26 / 43.0
3 / 29 / 50.4
4 / 26 / 25.1
5 / 29 / 61.1
6 / 26 / 54.9
7 / 30 / 28.9
8 / 27 / 12.7
Control group
9 / 29 / 20.0
10 / 25 / 14.4
11 / 29 / 43.8
12 / 29 / 28.2
13 / 30 / 27.7
Bournemouth
Experimental group
A / 71 (3 classes) / 18.1
B / 55 (2 classes / 27.5
C / 106 (4 classes) / 5.7
D / 25 / 41.2
Control group
E / 79 (3 classes) / 3.2
F / 88 (3 classes) / 19.9
G / 60 (2 classes) / 14.9
H / 80 (3 classes) / 9.2
Total (Experimentals) / 479
Total (Controls) / 449
Mean (all Controls) / 20.1
Mean (H&F Experimentals) / 37.0
Mean (Bournemouth Exps) / 23.1
H&F had been given by the Government in 1998 a ‘Single Regeneration Budget’—that is, a considerable sum of money in recognition of the social disadvantage in the population supplying the LEA schools’ intake. Some of it was spent in funding the previous CASE@KS1 project. The intention of the RCPCM project was to enhance the cognition and learning ability of the H&F pupils to at least the National average. At the same time it was thought desirable simultaneously to replicate the intervention in an LEA with a quite different social composition. Table 3 compares the ethnic compositions of the two LEAs.
Table 3: Ethnic composition of groups in this study
Ethnic groupEducation Authority / White% / Black% / Mixed% / Asian% / Chinese% / Other%
H&F
Experimental / 46.2 / 27.2 / 8.5 / 5.9 / 0.4 / 10.8
Control / 56.5 / 23.9 / 11.0 / 2.9 / 0.3 / 3.4
Bournemouth
Experimental / 96.4 / 0.5 / 1.5 / 0.2 / 0.2 / 0.6
Control / 89.3 / 0.5 / 5.6 / 2.1 / 0.3 / 1.0
The details in both Table 2 and Table 3 are of the whole schools in 2002/2003, so may not exactly describe the Y1 year-groups. The contrast between the two LEAs can clearly be seen in Table 3, but inspection of Table 2 shows there is socio-economic disadvantage comparable with that in H&F in the intake of at least two of the Bournemouth experimental schools.
The tests used
The Piagetian Spatial Relations test (NFER, 1979) was used in the original CSMS[5] research covering the range early concrete (2A) through the concrete generalisation (2B*) level. For the Y2 Post-test with the 7 year-olds it was found possible to use it in its original form, as Piaget’s original research featured children of this age. The two authors administered the test in H&F; in Bournemouth the first author tested the classes in one school as a demonstration for those testing the other schools. All the children’s responses are given in the form of their own drawings: no reading or writing is involved. The first question involves them drawing a jam-jar half-full of water at the various orientations the teacher shows them (upright, tilted at 30 degrees off vertical, and on its side: they have to imagine the water). The second gives them a drawn hill and asks them to draw a house on one side, and three trees on the other. The third repeats the agenda of question 1, but now with a jar containing a plumb-line hanging from its lid. The fourth, permitting an assessment at the 2B* level as well as all levels going down to Pre-operational, asks them to imagine themselves standing in the middle of a road consisting of an avenue of trees, and to draw what they would see, covering the near distance as well as afar. Assessment of the children’s responses consists of seeing how many aspects of the situation, how many relations, they can consider in the act of drawing. The scoring decisions are taken from the drawings in Piaget & Inhelder, 1956. Thus with the hill, if they can handle one relation at a time only (early concrete, 2A) then they relate the orientation of the house just to the slope of the hill. But if they can handle two relations (mature concrete, 2B) then they will also bear in mind earth’s horizontal, and they will draw the houses and the trees upright. In effect the test measures the complexity of relations the children can handle in the moment, and thus acts as a test of fluid intelligence (Shayer, 2008). The two authors assessed all the children’s drawings from both LEAs. The test reliability is 0.82. Items 1 and 3 have four scoring levels, ranging from <2A (early concrete) to 2B (mature concrete). Item 2 has five scoring levels, ranging from <2A to 2B* (concrete generalisation). Item 4 has six scoring levels, from <2A to the 2B*/3A (early formal) boundary. It is therefore equivalent to a 19 item test, and was scored using the rating scale Rasch analysis to give an equal interval scale from 2.79 (just below 3 = 2A, early concrete) to >6.3 (higher range of concrete generalisation).